Method for obtaining the absorption position, mass and rigidity of a particle

ABSTRACT

A method for obtaining the absorption position, mass and rigidity of a particle deposited on the surface of a resonator based on the relative change in the resonance frequency of said resonator in 3 or 4 flexural vibration modes. The rigidity of the particles is of great interest in the study of cells and other biological compounds that change state without significantly changing the mass.

FIELD OF THE INVENTION

The present invention relates to mass spectrometry, in particular to theuse of micro-cantilevers or bridges and their different vibration modesfor obtaining parameters of interest, such as mass.

BACKGROUND OF THE INVENTION

Mass spectrometry (MS) very accurately measures the mass-to-charge ratioin molecular species between 100 Da and 100 kDa. However, classicmethods do not offer sufficient efficiency with greater particle masses,such as cells, bacteria or viruses. New nanoelectromechanical systems(NEMS) such as cantilevers or bridges enable the mass of intact objectsgreater than 100 kDa to be measured, which means that these structuresare considered especially appropriate for studying biological complexesand nanoparticles. In nanoelectromechanical systems mass spectrometry,the sample is introduced by means of an electrospray ionization (ESI)system and the resulting ions are guided by means of an electrostaticsystem towards a high vacuum chamber (<; 10⁻⁵ Torr) where the resonatoris located. Alternatively, matrix-assisted laser desorption/ionization(MALDI) may be used to carry the sample to the resonator. As the sampleis absorbed by the resonator, sudden changes take place in the resonancefrequency thereof, changes that are proportional to the mass of saidparticle with a proportionality constant that depends on the absorptionposition. Given that resonance is independent of the charge of theparticle, the analysis of the data is simplified. The deconvolution ofthe absorption position throughout the NEMS and of the mass requiressimultaneous measurement of at least two vibration modes, as proposed byDohn et al. in “Mass and position determination of attached particles oncantilever based mass sensors”, Review of Scientific Instruments 78,103303, 2007) and is described in patent application US2014/0156224.However, these methods do not enable the rigidity to be measured, whichis a parameter that has been ignored to date as it is considered to haveno influence when calculating mass.

DESCRIPTION OF THE INVENTION

The present invention overcomes the technical prejudice described aboveand goes a step further in calculating parameters of interest byincluding the rigidity of particles, which is of great interest in thestudy of cells and other biological compounds that change state withoutsignificantly changing the mass (for example, healthy cells vs. cancercells).

As such, the invention consists of a method for obtaining the absorptionposition, mass and rigidity of a particle deposited on the surface of ananoelectromechanical resonator (cantilever or bridge) based on thechanges in frequency of 3 or 4 flexural vibration modes. The particlesmay be inorganic, virus, bacteria, protein or cell particles.

BRIEF DESCRIPTION OF THE FIGURES

In order to assist in a better understanding of the characteristics ofthe invention according to a preferred exemplary embodiment thereof andto complement this description, the following figures are attachedconstituting an integral part of the same, which, by way of illustrationand not limitation, represent the following:

FIG. 1 is a diagram of a mass spectrometry system that may be used tocarry out the method of the invention.

FIG. 2 shows the micro-cantilever in greater detail.

FIG. 3 is a schematic drawing of the beam deflection technique used tomeasure the resonance frequencies.

FIG. 4 is a graph showing the mass of particles taking into account therigidity compared to the mass calculated without taking into account therigidity and the mass provided by the manufacturer (continuous line).

FIG. 5 is a graph of the ration between the rigidity term and the massterm

$\left( {\epsilon = {\frac{\Delta_{s}}{\Delta_{m}} = \frac{\rho_{c}\chi_{e}E_{a}}{E_{c}\rho_{a}}}} \right)$of the particles.

DETAILED DESCRIPTION OF THE INVENTION

The method of the invention enables the absorption position ξ₀, the massterm

$\Delta_{m} = {\frac{1}{2}\frac{m_{a}}{m_{c}}}$and the rigidity term

$\Delta_{a} = {\frac{1}{2}\frac{V_{c}E_{a}}{V_{c}E_{c}}\chi_{a}}$of a particle deposited on the surface of a NEMS to be obtained based onthe measurement of the changes in frequency of 3 or 4 flexural vibrationmodes, where m is the mass, V is the volume, E is the Young module,χ_(e) is the number related to the form of adsorbate and how theadsorbate adheres to the resonator and the subscripts a and c refer tothe adsorbate and the resonator respectively. Therefore, knowing theproperties of the resonator, the mass m_(a) and the effective rigidityV_(a)E_(c)χ_(e) are obtained. This method enables the mass of adsorbatesto be calculated, for which the rigidity is important, with greateraccuracy.

The mass spectrometer (MS) used may be seen in FIG. 1. The MS is made upof three differentiated vacuum chambers. The first chamber (1) is atatmospheric pressure and connected to the second by means of a hotcapillary (8). The second chamber (2) is at 10 mbar and connected to thethird by means of a skimmer and a microhole (100 μm) (4). The thirdchamber is at 0.1 mbar and contains the nanoelectromechanical resonator(5), which may be a cantilever or bridge. In the section at atmosphericpressure there is an electrospray ionization system responsible forsending the sample to the sensors. The ESI is made up of a bottle (7)with the dissolution and particles (inorganic, viral, etc.) of interestin suspension and a polyether ether ketone (PEEK) capillary (6). Thedissolution is sent to the end of the capillary increasing the pressurein the bottle. When the first drop reaches the end of the capillary, ahigh voltage (3-5 kV) is applied, generating the known Taylor cone. Amicro-jet ejects from the point of the cone from which smallmicro-droplets emerge that are positively charged (if the high voltageis positive, and negative if the high voltage is negative), whichcontain the dissolution and particles of interest. The micro-dropletsgenerated in the first chamber (1) are sent to the second chamber (nextdifferential vacuum) (2) where there is the hot capillary (8). The innerdiameter of the hot capillary is between 400 and 500 μm and thetemperature thereof is fixed between 150 and 300°, which favours theevaporation of the dissolution and prevents the particles from stickingto the walls by means of thermal agitation. The positively chargeddroplets lose part of their mass but not their charge, which means thatthese are progressively divided into smaller drops. The breaking pointof a droplet into daughter droplets is caused when the Rayleighcriterion is reached, which is when the surface tension of the liquidand the repulsion caused by the positive charges (or negative in thecase that the high voltage is negative) are in equilibrium. Lastly, thedissolution evaporates completely, leaving only the particles ofinterest. The particles enter the third chamber (3), or thirddifferential vacuum, through a hole with a diameter of 100 μm. Thenanoelectromechanical resonator (5) is in the third vacuum differential.In a particular implementation, the resonator is a bridge or cantileveras shown in FIG. 2. The third chamber (3) has two optical windows thatenable the natural frequency of the resonator oscillation to be measuredusing the beam deflection method (FIG. 3). The laser (10) focuses on theresonator in the area where the product of the slopes of the vibrationmodes used is at a maximum by means of micro-positioners XYZ and thebeam reflected is collected by a photodetector (9). The cantilever isexcited by a piezoelectric element. When a particle reaches the surfaceof the resonator, there is a change in the frequency corresponding toeach vibration mode. Based on the relative frequency changes, the valuesof ξ₀, Δ_(m) and Δ_(s) are found by numerical calculation, whichmaximise the probability density function.

${{{JPDF}\left( \hat{\Omega} \right)} = \frac{e^{- \frac{{({\hat{\Omega} - M})}\Sigma^{- 1}\;{({\hat{\Omega} - M})}^{T}}{2}}}{\left( {2\pi} \right)^{N/2}\sqrt{\Sigma }}}\;$

The change in vibration frequency may be measured in several ways. In apreferred example, a LASER is focused on the resonator in the area wherethe product of the slope of the vibration modes used is maximised. Thereflected beam is detected by a 4 quadrant photo detector (or similarphotodetector), which is known in the state of the art of beamdeflection. Then, the signal from the photodetector is pre-amplified andsent to an amplifier, preferably of the Lock-in type (a type ofamplifier that can extract signals from incredibly noisy media). Apiezoelectric material located below the resonator is used to carry outa sweep around the frequencies of interest in order to obtain thecharacteristic frequencies and phases of the resonator. The frequenciesand phases obtained in the point about are used to configure the phaselock loops (PLLs), which monitor the corresponding frequencies overtime. When a particle reaches the surface of the resonator, there is achange in the resonance frequencies Δf_(n). This change is registered bythe PLLs. The changes in frequency are stored and, based on the same,the changes in relative frequency are calculated using the followingformula:

$\mu_{n} = {\frac{\Delta\; f_{n}}{f_{0n}} = \frac{f_{n} - f_{0n}}{f_{0n}}}$

Where f_(n) is the average of the frequency over the time correspondingto the mode n after absorption and f_(0n) is the average of thefrequency over the time corresponding to the mode n before absorption.In this way, the relative changes of each vibration mode are obtaineddepending on the time and the standard deviation thereof.

In order to obtain the absorption position, mass and rigidity of theadsorbed particle based on these data, which have been stored, thefollowing steps must be carried out:

1. The standard deviation of the relative frequency change of each modeσ_(n) and the values of the relative frequency change corresponding toadsorption μ_(n) is calculated based on the data stored.

2. The following probability density function is formed based on thevalues of μ_(n) and σ_(n) for the N modes used (N=3 or N=4) which dependon three variables ξ₀, Δ_(m) and Δ_(s);

${{JPDF}\left( \hat{\Omega} \right)} = \frac{e^{- \frac{{({\hat{\Omega} - M})}{\Sigma^{- 1}{({\hat{\Omega} - M})}}^{T}}{2}}}{\left( {2\pi} \right)^{N/2}\sqrt{\Sigma }}$

Where {circumflex over (Ω)}=(Ω₁, Ω₂, . . . , Ω_(N)), with Ω_(a) givenby:

$\Omega_{n} = {{{- \Delta_{m}}{\psi_{n}\left( \xi_{0} \right)}^{2}} + {\Delta_{s}\frac{1}{\beta_{n}^{4}}\left( \frac{d^{2}{\psi_{n}\left( \xi_{0} \right)}}{d\;\xi^{2}} \right)^{2}}}$

Where ψ_(n) and β_(n) are the type of vibration and the eigenvalue ofthe n-th mode respectively,

M−(μ₁, μ₂, . . . , μ_(N)) and Σ is the covariance matrix given by:

$\Sigma = \begin{pmatrix}\sigma_{1}^{2} & {\sigma_{1}\sigma_{2\;}\rho_{12}} & \ldots & {\sigma_{1}\sigma_{N}\rho_{1N}} \\{\sigma_{1}\sigma_{2}\rho_{12}} & \sigma_{2}^{2} & \ldots & \vdots \\\vdots & \vdots & \ddots & \vdots \\{\sigma_{1}\sigma_{N}\rho_{1N}} & \ldots & \ldots & \sigma_{N}^{2}\end{pmatrix}$

Where ρ_(ij) is the correlation between modes i and j.

3. The values of ξ₀, Λ_(m) and Λ_(s) that maximise the probabilitydensity function JPDF({circumflex over (Ω)}) is found. At this point, aperson skilled in the art will recognise that there are several methodsfor obtaining the values of ξ₀, Λ_(m) and Λ_(s) that maximise theprobability density function JPDF({circumflex over (Ω)}). Two of themare proposed below.

First Exemplary Embodiment

The following functional is formed:F=({circumflex over (Ω)}(Δ_(m),Δ_(s),ξ₀)−M)Σ⁻¹({circumflex over(Ω)}(Δ_(m),Δ_(s),ξ₀)−M)^(T)

Functional F is numerically minimised using any existing optimisationroutine, for example Newton's method.

Second Exemplary Embodiment

The following functional is formed:

${G\left( {\xi_{0},\epsilon} \right)} = {\sum\limits_{n = 1}^{N}\left( {{C_{n}\left( {\xi_{0},\epsilon} \right)} - \frac{\mu_{n}}{\sqrt{\sum\limits_{m = 1}^{N}\mu_{m}^{2}}}} \right)^{2}}$

Where

$\epsilon = \frac{\Delta_{g}}{\Delta_{m}}$and C_(n)(ξ₀,∈) are given by

$C_{n} = \frac{\Omega_{n}\left( {\xi_{0},\epsilon} \right)}{\sqrt{\sum\limits_{m = 1}^{N}{\Omega_{m}\left( {\xi_{0},\epsilon} \right)}^{2}}}$

Where Ω_(n)(ξ₀, ∈) is given by:

${\Omega_{n}\left( {\xi_{0},\epsilon} \right)} = {\Delta_{m}\left( {{- {\psi_{n}\left( \xi_{0} \right)}^{2}} + {\epsilon\frac{1}{\beta_{n}^{4}}\left( \frac{d^{2}{\psi_{n}\left( \xi_{0} \right)}}{d\;\xi^{2}} \right)^{2}}} \right)}$

The values of ξ₀ and ∈ that minimise the functional G are found usingany existing numerical routine (again, Newton's method can be used).

The following functional is formed:F=({circumflex over (Ω)}(Δ_(m),∈,ξ₀)−M)Σ⁻¹({circumflex over(Ω)}(Δ_(m),∈,ξ₀)−M)^(T)

The previously obtained values of ξ₀ and ∈ are used and are substitutedin the functional F.

The value of that minimised the functional F is found. Therefore, ξ₀,Δ_(m) and Δ_(s)=∈Δ_(m) are perfectly determined. This method hascomputational advantages with respect to the first due to the fact thatthe function to be minimised has two variables instead of three. As wellas these computational advantages, this method is also more accuratethan the first.

Examples

FIG. 4 is obtained by following the second method for gold nanoparticlesof 100 nm with a nominal diameter. 174 absorptions of said nanoparticleswere measured on the surface of a cantilever. Frequency changes in thefirst 3 flexural modes were stored and the second method described abovewas used to extract the mass, position and rigidity of the goldnanoparticles. A graph of the mass (FIG. 4) and another for ∈) (FIG. 5)was prepared using this data. The graph of the mass also shows thedistribution provided by the manufacturer (Sigma Aldrich, segment withcontinuous line) and the distribution of the mass obtained withouttaking into account the rigidity (method described in the state of theart) where a displacement in the distribution of the mass towardssmaller values when rigidity is not taken into account may be clearlyseen. The graph also shows that the data that include the rigidityeffect fit better with the distribution of the mass provided by themanufacturer, such that the method of the invention presents theadditional advantage of improving the measurements of the massparameter.

Based on the definitions provided above for Δ_(m) and Δ_(s), theparameter ∈ may be expressed as:

$\epsilon = {\frac{\Delta_{s}}{\Delta_{m}} = \frac{\rho_{c}\chi_{e}E_{a}}{E_{c}\rho_{a}}}$

Where ∈ is a direct measurement and is proportional to the rigidity ofthe adsorbate.

The invention claimed:
 1. A method for obtaining the absorptionposition, mass and rigidity of a particle deposited on the surface of aresonator of a mass spectrometer based on the relative change in theresonance frequency of said resonator in N=3 or N=4 flexural vibrationmodes, where said method comprises the following steps: a. the standarddeviation of the relative frequency change of each mode σ_(n) and thevalues of the relative frequency change corresponding to adsorptionμ_(n) is calculated, b. the following probability density function isformed${{JPDF}\left( \hat{\Omega} \right)} = \frac{e^{- \frac{{({\hat{\Omega} - M})}{\Sigma^{- 1}{({\hat{\Omega} - M})}}^{T}}{2}}}{\left( {2\;\pi} \right)^{N/2}\sqrt{\Sigma }}$where {circumflex over (Ω)}=(Ω₁, Ω₂, . . . , Ω_(N)), with Ω_(n) givenby:$\Omega_{n} = {{{- \Delta_{m}}{\psi_{n}\left( \xi_{0} \right)}^{2}} + {\Delta_{s}\frac{1}{\beta_{n}^{4}}\left( \frac{d^{2}{\psi_{n}\left( \xi_{0} \right)}}{d\;\xi^{2}} \right)^{2}}}$where ψ_(n) and β_(n) are the type of vibration and the eigenvalue ofthe n-th mode, respectively, M=(μ₁, μ₂, . . . , μ_(N)) and Σ is thecovariance matrix given by: $\Sigma = \begin{pmatrix}\sigma_{1}^{2} & {\sigma_{1}\sigma_{2}\rho_{12}} & \ldots & {\sigma_{1}\sigma_{N}\rho_{1\; N}} \\{\sigma_{1}\sigma_{2}\rho_{12}} & \sigma_{2}^{2} & \ldots & \vdots \\\vdots & \vdots & \ddots & \vdots \\{\sigma_{1}\sigma_{N}\rho_{1\; N}} & \ldots & \ldots & \sigma_{N}^{2}\end{pmatrix}$ where ρ_(ij) is the correlation between modes i and j; c.the values of the absorption position ξ₀, the mass term Δ_(m) and therigidity term Δ_(s) that maximise the probability density function arecalculated.
 2. The method of claim 1, wherein the particles areinorganic, virus, bacteria, protein or cell particles.
 3. The methodaccording to claim 1, wherein, in order to maximise the probabilitydensity function JPDF({circumflex over (Ω)}), the following functionalis minimised:F=({circumflex over (Ω)}(Δ_(m),∈,ξ₀)−M)Σ⁻¹({circumflex over(Ω)}(Δ_(m),∈,ξ₀)−M)^(T) where$\epsilon = {\frac{\Delta_{s}}{\Delta_{m}}.}$
 4. The method of claim 3,wherein the particles are inorganic, virus, bacteria, protein or cellparticles.
 5. The method according to claim 1, wherein, in order tomaximise the probability density function JPDF({circumflex over (Ω)}),the following functional is formed:${G\left( {\xi_{0},\epsilon} \right)} = {\sum\limits_{n = 1}^{N}\left( {{C_{n}\left( {\xi_{0},\epsilon} \right)} - \frac{\mu_{n}}{\sqrt{\sum\limits_{m = 1}^{N}\mu_{m}^{2}}}} \right)^{2}}$where $\epsilon = \frac{\Delta_{g}}{\Delta_{m}}$ and C_(n)(ξ₀,∈) aregiven by:${C_{n} = \frac{\Omega_{n}\left( {\xi_{0},\epsilon} \right)}{\sqrt{\sum\limits_{m = 1}^{N}{\Omega_{m}\left( {\xi_{0},\epsilon} \right)}^{2}}}},$where Ω_(n)(ξ₀,∈) is given by:${{\Omega_{n}\left( {\xi_{0},\epsilon} \right)} = {\Delta_{m}\left( {{- {\psi_{n}\left( \xi_{0} \right)}^{2}} + {\epsilon\frac{1}{\beta_{n}^{4}}\left( \frac{d^{2}{\psi_{n}\left( \xi_{0} \right)}}{d\;\xi^{2}} \right)^{2}}} \right)}},$the values of ξ₀ and ∈ that minimise the functional G are found usingany existing numerical routine, the following functional is formed:F=({circumflex over (Ω)}(Δ_(m),∈,ξ₀)−M)Σ⁻¹({circumflex over(Ω)}(Δ_(m),∈,ξ₀)−M)^(T) the previously obtained values of ξ₀ and ∈ areused, they are substituted in the functional F, the value of Δ_(m) thatminimises the functional F is found.
 6. The method of claim 5, whereinthe particles are inorganic, virus, bacteria, protein or cell particles.7. The method according to claim 1, wherein the resonator is acantilever or a bridge.
 8. The method according to claim 7, wherein, inorder to maximise the probability density function JPDF({circumflex over(Ω)}), the following functional is minimised:F=({circumflex over (Ω)}(Δ_(m),∈,ξ₀)−M)Σ⁻¹({circumflex over(Ω)}(Δ_(m),∈,ξ₀)−M)^(T) where ∈=Δs/Δm
 9. The method of claim 8, whereinthe particles are inorganic, virus, bacteria, protein or cell particles.10. The method according to claim 7, wherein, in order to maximise theprobability density function JPDF({circumflex over (Ω)}), the followingfunctional is formed:${G\left( {\xi_{0},\epsilon} \right)} = {\sum\limits_{n = 1}^{N}\left( {{C_{n}\left( {\xi_{0},\epsilon} \right)} - \frac{\mu_{n}}{\sqrt{\sum\limits_{m = 1}^{N}\mu_{m}^{2}}}} \right)^{2}}$where $\epsilon = \frac{\Delta_{g}}{\Delta_{m}}$ and C_(n)(ξ₀,∈) aregiven by:${C_{n} = \frac{\Omega_{n}\left( {\xi_{0},\epsilon} \right)}{\sqrt{\sum\limits_{m = 1}^{N}{\Omega_{m}\left( {\xi_{0},\epsilon} \right)}^{2}}}},$where Ω_(n)(ξ₀,∈) is given by:${{\Omega_{n}\left( {\xi_{0},\epsilon} \right)} = {\Delta_{m}\left( {{- {\psi_{n}\left( \xi_{0} \right)}^{2}} + {\epsilon\frac{1}{\beta_{n}^{4}}\left( \frac{d^{2}{\psi_{n}\left( \xi_{0} \right)}}{d\;\xi^{2}} \right)^{2}}} \right)}},$the values of ξ₀ and ∈ that minimise the functional G are found usingany existing numerical routine, the following functional is formed:F=({circumflex over (Ω)}(Δ_(m),∈,ξ₀)−M)Σ⁻¹({circumflex over(Ω)}(Δ_(m),∈,ξ₀)−M)^(T) the previously obtained values of ξ₀ and ∈ areused, they are substituted in the functional F, the value of Δ_(m) thatminimises the functional F is found.
 11. The method of claim 10, whereinthe particles are inorganic, virus, bacteria, protein or cell particles.12. The method of claim 7, wherein the particles are inorganic, virus,bacteria, protein or cell particles.